Question

if sn denotes the sum of first n terms of an AP , prove that S12 = 3(S8-S4)​

Answer 1

Let 1  

st

 term of A.P=a

and common difference =d

We have to prove that

S  

12

​

=3(S  

8

​

−S  

4

​

)

S  

12

​

=  

2

12

​

[2a+(12−1)d]

S  

8

​

=  

2

8

​

[2a+(8−1)d]

S  

4

​

=  

2

4

​

[2a+(4−1)d]

Now, R.H.S=3(S  

8

​

−S  

4

​

)

=3[  

2

8

​

(2a+(8−1)d−  

2

4

​

(2a+(4−1)d]

=3[8x+28d−4a−6d]

=6(2a+11d)

=  

2

12

​

[2a+(12−1)d]

=S  

12

​

 

=L.H.S